A LINEAR PROGRAMMING TO PLANNING PRODUCTION IN SWINE FARM Sara V. - - PowerPoint PPT Presentation

Sara V. Rodríguez, Luis M. Plá Departament of Mathematics, University of Lleida, Spain Victor Albornoz Departament of Industrias, Universidad Técnica Federico Santa María, Chile

A LINEAR PROGRAMMING TO PLANNING PRODUCTION IN SWINE FARM

Outline

• 1. Introduction
• Overview of swine industry
• Swine production system
• Literature review
• 2. Description problem
• Lifespand of the sow
• Reproductive cycle
• 3. Linear programming model
• Formulation of the problem
• Numerical test
• Sensitivity analysis
• Dealing with the weakness of LP
• Stochastic model extension
• 4. Further research and conclusions

UNCERTAINTY DECISION VARIABLES CONSTRAINTS

Overview to Swine Industry

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Probabilistic System

PROFITABILITY MARGIN

In Europe the decreasing in the profit margin during the last years has motivated the use of operation reseach methods to make better decisions.

Overview to Swine Industry

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Swine Production System

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Danish Swine Production System

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Spanish Production System

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Literature Review

• Strategic decisions
• Infinite horizon

Models

• Simulation
• Optimization
• Dynamic Programming; Markov models

Relevant aspects:

Introduction Description problem LP model Stochastic programming Further research

Pla, L.M., 2007. Review of mathematical models for sow herd

• management. Livestock Science 106, 107–119.

Introduction Description problem LP model Stochastic programming Further research

SIMULATION MODELS

• Allen, M.A., Stewart, T.S., 1983. A simulation model for a swine breeding

unit producing feeder pigs. Agricultural Systems 10, 193–211.

• Marsh, W.E., 1986. Economic decision making on health and management

livestock herds: examining complex problems through computer simulation. Ph.D. thesis, University of Minnesota, St. Paul.

• Jalving, A.W., 1992. The possible role of existing models in on-farm decision

support in dairy cattle and swine production. Livestock Production Science 31, 351–365.

• Plà, L.M., Pomar, C., Pomar, J., 2003. A Markov decision sow model

representing the productive lifespan of sows. Agricultural Systems76, 253–272.

OPTIMIZATION MODELS

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• Dijkhuizen, A.A., Morris, R.S., Morrow, M., 1986. Economic optimisation of

culling strategies in swine breeding herds, using the “PORKCHOP computer program”. Preventive Veterinary Medicine 4, 341–353.

• Huirne, R.B., Dijkhuizen, A.A., Van Beek, P., Hendriks, Th.H.B., 1993.

Stochastic dynamic programming to support sow replacement decisions. European Journal of Operational Research 67, 161–171.

• Kristensen, A.R., Søllested, T.A., 2004b. A sow replacement model using

Bayesian updating in a three-level hierarchic Markov process II. Optimization

• model. Livestock Production Sciences 87, 25–36.

Aim

a) Formulate a linear optimization model. b) Formulate a linear stochastic extension. Tactical decisions Linear Finite horizon Stochastic two stage.

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Swo farm

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INSEMINATION GESTATION LACTATION ENTRY CYCLE NEXT CYCLE

Swine Production System

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Identification of the problem

• How many sows the farmer should replace?
• How many sows the farmer should buy?
• How many insemintions the farmer should accept?
• What is the optimal cycle in which the sow should be replace?
• What is the optimal reproductive state in which the sow should be

replace?

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Linear Programming Model

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Linear Programming Model

• Constraints

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Reference Dantzig, G.B. y Thapa, M.N., 1996. Linear Programming. Introduction. Springer Series in Operations Research, Springer. Max cTx s.a. Ax ≤ b x ≥ 0

• Objective Function

L 1 L 3 L 4 L 2

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WEEK

LACTATION

I 1

WEEK

INSEMINATION

PERIOD= WEEK

G 15 G 16 ZR 1 ZR 3 G 4 ZR 2

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WEEK

GESTATION BREEDING- CONTROL

GESTATION

r ={1,2,3} k ={1,2,3}

ZR 1 Z G 15 G 16 L 1 L 3 L 4 L 2 ZR 1 Z G 15 G 16 L 1 L 3 L 4 L 2

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PERIOD= WEEK

C={1, 2, 3, ...} Set of number of cycles. Sg={1, 2, ..., 16} Set of number of gestation week Sl={1, 2, 3, 4} Set of number of lactation week T={1, ... 52} Set of periods Nr={1,..3} Set of repetitions of insemination Sr={1,..,3} Set of insemination waiting week α(t,c,g) = Survival rate of gestation of the period t, cycle c and gestation week g. β(t,c,r) = Survival rate of insemination of the period t, cycle c, waiting the reinsemination r. γ(t,c) = Number of weaned piglets at period t and cycle c.

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Y(t,c,l) Number of sows in lactation state at period t, cycle c, lactation week l. X(t,c,g) Number of sows in gestation state at period t, cycle c, gestation week g. Z(t,c) Number of sows in insemination state at period t, cycle c. ZR(t,c,r,k) Numer of sows in breedig-control state at period t, cycle c, waiting the reinsemination r, waiting week k. UL(t,c) Number of replaced sows at the end of lactation state. UZ(t,c,r) Number of replaced sows at the end of insemination state, waiting the reinsemination r.

Introduction Description problem LP model Stochastic programming Further research

∑∑ ∑ ∑∑∑ ∑∑ ∑

∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

+ + +

T t c t C c t T t T t C c r c t Nr r t T t C c c t t C c l c t c t c t

AB ru UZ ru UL ru Y rl

, , , , , , , ,

* * * * *

*

γ

Objective Function

Introduction Description problem LP model Stochastic programming Further research

Objective Function

Introduction Description problem LP model Stochastic programming Further research

Objective Function

Introduction Description problem LP model Stochastic programming Further research

INSEMINATION GESTATION LACTATION BREEDING- CONTROL

l c t T t C c Sl l l c t g c t T t C c Sr Sg g g c t T t C c k r c t Nr r Sr k k c t T t C c c t c t

Y rl X rx ZR rzr Z rz

, , , , , , , , , , , , , , , ,

* * * *

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

∈ ∈ ∈ ∈ ∈ − ∈ ∈ ∈ ∈ ∈ ∈ ∈

+ + +

rz t,1

Objective Function

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l c t T t C c Sl l l c t g c t T t C c Sr Sg g g c t T t C c k r c t Nr r Sr k k c t T t C c c t c t

Y rl X rx ZR rzr Z rz

, , , , , , , , , , , , , , , ,

* * * *

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

∈ ∈ ∈ ∈ ∈ − ∈ ∈ ∈ ∈ ∈ ∈ ∈

+ + +

∑∑ ∑ ∑∑ ∑ ∑∑ ∑

∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

+ + +

T t c t C c t T t T t C c r c t Nr r t T t C c c t t C c l c t c t c t

AB ru UZ ru UL ru Y rl

, , , , , , , ,

* * * * *

*

γ

) 4 ( ) 3 ( ) 2 ( ) 1 ( } 1 {

, , , 1 , , , 1 , , , , , 1 , 1

… … … … C c Sl l Y Y C c Sg g X X C c Sr k Nr r ZR Z C c Z Z

l c l c g c g c k r c k r c c c

∈ ∈ = ∈ ∈ = ∈ ∈ ∈ = − ∈ =

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Constraints

) 5 ( } 1 { } 1 {

1 , 1 , 1 , 1 ,

*

… − ∈ − ∈ − =

− − − −

C c T t UL Y Z

c t l c t c t

LACTATION 4 INSEMINATION Z

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) 8 ( } 1 { } 1 { ) 7 ( } 1 { } 1 { ) 1 ( ) 6 ( } 1 {

1 , , , 1 , , , 1 , , 1 3 , 1 , , 1 1 , , 1 1 , , , , , 1 1 , 1 , ,

… … … − ∈ ∈ ∈ − ∈ = − ∈ ∈ − ∈ − − = ∈ − ∈ =

− − − − − − − − −

Sr k Nr r C c T t ZR ZR Nr r C c T t UZ ZR ZR C c T t Z ZR

k r c t k r c t r c t r c t r c t r c t c t c t

β ZR 1 Z ZR 1 ZR 2 G ZR 2 ZR 3 ZR 3

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UR

1-β(t,c,r)

G 4 ZR G 1 G 16 G 4 G 15 G 16

) 10 ( } 1 { } 1 { ) 9 ( } 1 {

1 , , 1 1 , , , 3 , , , 1 , , 1 4 , ,

… … − − ∈ ∈ − ∈ = ∈ − ∈ =

− − − − ∈ −

∑

Sr Sg g C c T t X X C c T t ZR X

g c t g c g c t r c t Nr r r c t c t

α β

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G 16 L 1 L 3 L 4 L 2 L 1 L 3 L 4 L 2

) 12 ( } 1 { } 1 { ) 11 ( } 1 {

1 , , 1 , , , , 1 , 1 1 , ,

* *

… … − ∈ ∈ − ∈ = ∈ − ∈ =

− − − −

Sl l C c T t Y Y C c T t X Y

l c t l c t g c t g c t c t

α

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) 15 ( ) 14 ( ) 13 (

, , , , } { , , , , , ,

* *

… … … T t CL Y X T t CG X T t CI ZR Z

C c L l l c t C c g c t C c g Sr Sg g g c t C c Nr r Sr k k r c t C c c t

∈ ≤ + ∈ ≤ ∈ ≤ +

∑∑ ∑ ∑ ∑ ∑∑∑ ∑

∈ ∈ ∈ ∈ − − ∈ ∈ ∈ ∈ ∈

Introduction Description problem LP model Stochastic programming Further research

BD-Porc databank [http://www.irta.es/bdporc/ OPL Development Studio Version 5

Introduction Description problem LP model Stochastic programming Further research

Maximun cycle: 8 Finit horizon: 52 weeks Initial conditions:

Numerical Test Results

• Objective function = 9,4781 e+5

500 1000 1500 2000 2500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

The behaviour of the Z(t,1)

Period

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50 100 150 200 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

The size of the herd

Period

) 21 ( ) 20 ( ) 19 ( ) 18 (

, , , , , , , ,

* * * *

… … … …

∑ ∑ ∑ ∑ ∑

∈ ∈ ∈ ∈ ∈

∈ ≥ − ∈ ≥ ∈ ≥ ≥

C c l c t C c g c t k C c NR r k r c t C c c t

Sl l yf Y Sr Sg g xf X Sr k zrf ZR zf Z

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Introduction Description problem LP model Stochastic programming Further research

500 1000 1500 2000 2500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

The size of the herd without FSC

500 1000 1500 2000 2500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

The size of the herd with FSC

• OF= 9,48 e+5
• OF= 6,11 e+5

) 17 ( } { 40 20 ) 16 ( } { 10

* 1 , * 1 , 1 , 1

… … t T t Z t T t Z Z

t t t

− ∈ ≤ ≤ − ∈ ≤ −

+

What happen if we use purchase constraints?

Period

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The behaviour of the Z(t,1) without purchase constraints

Period

20 40 60 80 100 120 140 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 20 40 60 80 100 120 140 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

• OF= 6,11 e+5
• OF= 6,03 e+5

( )

T t t d c t Y c t

N c

∈ ≤

∑

∈

) ( ) 4 , , ( , γ

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• 10

10 30 50 70 90 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

1900 2000 2100 2200 2300

The behaviour of the Z(t,1) and the size of the sow herd.

Period

Size of the herd Z(t,1)

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Sensitive Analysis (data)

How sensitive is the optimal value when we change population dynamics parameters?. α(t,c,g) , β(t,c,r) , γ(t,c) α(t,c,g) β(t,c,r) γ(t,c) Model 1 Model 2 Model 3 Base Base Base +5% +5% +5%

• 5%
• 5%
• 5%

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Sensitive Analysis (result)

How sensitive is the optimal value when we change population dynamics parameters?.

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100000 200000 300000 400000 500000 600000 700000 800000 900000 1 2 3

Modelo2 Modelo1 Modelo3

• 35%
• 35%

Modelo2 Modelo1 Modelo3 OF 3.9345e+5 6.0341e+5 8.1640e+5

Dealing with the weakness of LP

Introduction Description problem LP model Stochastic programming Further research

Kristensen A.R.,,2008. Advanced Herd Management.

Stochastic programming

Introduction Description problem LP model Stochastic programming Further research

The stochastic programming models can be considered as a extension of the linear optimization models where now the parameters are given in probabilistic terms. Historically the following autors and articles can be cited as the begining

• f this area.
• G.Dantzig, 1955. Linear Programming Under Uncertainty. Management

Science 1 (3), 197-206.

• A.Charnes and W.W. Cooper, 1959. Chance-Constrained Programming.

Management Science 6 (1).

• A. Madansky, 1960. Inequalities for Stochastic Linear Programming

Management Science 6, 197-204.

Two Steps Stochastic Model

Here and now Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario |Ω| Wait and see

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Reference Birge J.R. and Louveaux F., 2002. Introduction to Stochastic Programing, Springer.

The main sources of uncertainty observed in the previous model were precisely the population dynamics parameters. ω(t,c,s) = Survival rate of insemination at period t, cycle c and scenario s. α(t,c,g,s) = Survival rate of gestation at period t, cycle c, gestation week g and scenario s. γ(t,c,s) = Number of weaned piglets at period t, cycle c and scenario s.

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Two Steps Stochastic Model

BD-Porc databank [http://www.irta.es/bdporc/ OPL Development Studio Version 5

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α(t,c,g) β(t,c,r) γ(t,c) Scenary 1 Scenary 2 Scenary 3 Base Base Base +5% +5% +5%

• 5%
• 5%
• 5%

Stochastic model

Stochastic Programming Model

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Numerical Test Results

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Modelo2 Modelo1 Modelo3 OF 3.9345e+5 6.0341e+5 8.1640e+5 Stochastic 5.8461e+5

Value of information and Stochastic Soluciton

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Modelo2 Modelo1 Modelo3 OF 3.9345e+5 6.0341e+5 8.1640e+5

EVPI= VSS=

Stochastic 5.8461e+5 6.0442e+5 1.981e+4

∑

Ω ∈

=

s s IP s Z

w EEV1

VSS = EEV2-ZIP.

s IP

Z

VSS is the value of the Stochastic solution EEV1 is the expected value being the value of each deterministic model.

Further Research and Conclusions

• The proposed models seem to be a suitable and valuable

methodology for supporting the management of sow farms.

• All in all, the model characterizes significant tactical

decisions from commonly- implemented industry rules-of thumb and possesses the potencial to increase profitability in and industry characterized by narrow profit margins.

• Generating scenarios for one year period, ideally to be

robust to different scenario sets.

• Under a more complicated scenario tree we need to study a

multi-stage stochastic program.

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THANKS FOR YOUR ATTENTION